Soft Computing

, Volume 21, Issue 8, pp 1923–1936 | Cite as

A novel soft rough fuzzy set: Z-soft rough fuzzy ideals of hemirings and corresponding decision making

Foundations

Abstract

This paper introduces the notion of Z-soft rough fuzzy sets of hemirings, which is an extended notion of soft rough sets and rough fuzzy sets. It is pointed out that this novel concept removes the limiting condition that full soft sets require in Feng-soft rough fuzzy sets and Meng-soft rough fuzzy sets. We study roughness in hemirings with respect to a ZS-approximation space. Some new soft rough fuzzy operations over hemirings are explored. In particular, Z-lower and Z-upper soft rough fuzzy ideals (k-ideals, h-ideals, strong h-ideals) are investigated. Finally, we put forth an approach for decision making problem based on Z-soft rough fuzzy sets and give an example. Corresponding decision making methods based on Z-soft rough fuzzy sets are analysed.

Keywords

Soft rough fuzzy set Z-soft rough fuzzy set Z-soft rough fuzzy ideal Decision making 

References

  1. Aktaş H, Çaǧman N (2007) Soft sets and soft groups. Inf Sci 177(13):2726–2735MathSciNetCrossRefMATHGoogle Scholar
  2. Ali MI (2011) A note on soft sets, rough soft sets and fuzzy soft sets. Appl Soft Comput 11:3329–3332CrossRefGoogle Scholar
  3. Ali MI, Feng F, Liu XY, Min WK, Shabir M (2009) On some new operations in soft set theory. Comput Math Appl 57(9):1547–1553MathSciNetCrossRefMATHGoogle Scholar
  4. Çaǧman N, Enginoǧlu S (2010) Soft set theory and uni-int decision making. Eur J Oper Res 207(2):848–855MathSciNetCrossRefGoogle Scholar
  5. Chen D, Tsang ECC, Yeung DS, Wang X (2005) The parameterization reduction of soft sets and its applications. Comput Math Appl 49:757–763MathSciNetCrossRefMATHGoogle Scholar
  6. Feng F, Li C, Davvaz B, Ali MI (2010) Soft sets combined with fuzzy sets and rough sets: a tentative approach. Soft Comput 14(9):899–911CrossRefMATHGoogle Scholar
  7. Feng F, Liu X, Leoreanu-Fotea V, Jun YB (2011) Soft sets and soft rough sets. Inform Sci 181:1125–1137MathSciNetCrossRefMATHGoogle Scholar
  8. Feng F, Li Y, Leoreanu-Fotea V (2010) Application of level soft sets in decision making based on interval-valued fuzzy soft sets. J Comput Math Appl 60(6):1756–1767MathSciNetCrossRefMATHGoogle Scholar
  9. Jun YB (2008) Soft \(BCK/BCI\)-algebras. Comput Math Appl 56(5):1408–1413MathSciNetCrossRefMATHGoogle Scholar
  10. Kong Z, Zhang G, Wang L (2014) An efficient decision making approach in incomplete soft set. Appl Math Model 38(7):2141–2150MathSciNetCrossRefGoogle Scholar
  11. Kong Z, Gao LQ, Wang LF (2007) Comment on a fuzzy soft set theoretic approach to decision making problems. J Comput Math Appl 223:540–542CrossRefMATHGoogle Scholar
  12. Li Z, Wen G, Han Y (2014) Decision making based on intuitionistic fuzzy soft sets and its algorithm. J Comput Anal Appl 17(4):620–631MathSciNetMATHGoogle Scholar
  13. Li Z, Wen G, Xie N (2015) An approach to fuzzy soft sets in decision making based on grey relational analysis and Dempster-Shafer theory of evidence: An application in medical diagnosis. Artif Intell Med 64:161–171CrossRefGoogle Scholar
  14. Li Z, Xie T (2014) The relationship among soft sets, soft rough sets and topologies. Soft Comput 18:717–728CrossRefMATHGoogle Scholar
  15. Li Z, Xie T (2015) Roughness of fuzzy soft sets and related results. Int J Comput Intell Syst 8:278–296CrossRefGoogle Scholar
  16. Li Z, Xie N, Wen G (2015) Soft coverings and their parameter reductions. Appl Soft Comput 31:48–60CrossRefGoogle Scholar
  17. Ma X, Zhan J (2015) Applications of soft intersection set theory to \(h\)-hemiregular and \(h\)-semisimple hemirings. J Mult Valued Logic Soft Comput 25:105–124MathSciNetGoogle Scholar
  18. Ma X, Zhan J (2014) Applications of a new soft set to \(h\)-hemiregular hemirings via \((M, N)-SI-h\)-ideals. J Intell Fuzzy Syst 26:2515–2525MathSciNetMATHGoogle Scholar
  19. Maji PK, Roy AR (2002) An application of soft sets in a decision making problem. Comput Math Appl 44:1077–1083MathSciNetCrossRefMATHGoogle Scholar
  20. Meng D, Zhang X, Qin K (2011) Soft rough fuzzy sets and soft fuzzy rough sets. Comput Math Appl 62(12):4635–4645MathSciNetCrossRefMATHGoogle Scholar
  21. Molodtsov D (1999) Soft set theory-first results. Comput Math Appl 37(4–5):19–31MathSciNetCrossRefMATHGoogle Scholar
  22. Molodtsov D (2004) The Theory of Soft Sets. URSS Publishers, Moscow (in Russian)Google Scholar
  23. Pawlak Z (1982) Rough sets. Int J Inf Comp Sci 115:341–356CrossRefMATHGoogle Scholar
  24. Roy AR, Maji PK (2007) A fuzzy soft set theoretic approach to decision making problems. J Comput Math Appl 203(2):412–418CrossRefMATHGoogle Scholar
  25. Shabir M, Ali MI, Shaheen T (2013) Another approach to soft rough sets. Knowl Based Syst 40(1):72–80CrossRefGoogle Scholar
  26. Sun B, Ma W (2014) Soft fuzzy rough sets and its application in decision making. Artif Intell Rev 41(1):67–80CrossRefGoogle Scholar
  27. Sun B, Ma W (2013) An approach to decision making based on intuitionistic fuzzy rough set over two universes. J Oper Res Soc 64(7):1079–1089CrossRefGoogle Scholar
  28. Sun B, Ma W (2011) Fuzzy rough set model on two different universed and its application. Appl Math Model 35(4):1798–1809MathSciNetCrossRefMATHGoogle Scholar
  29. Sun B, Ma W, Chen D (2014) Rough approximation of a fuzzy concept on a hybrid attribute information system and its uncertainty measure. Inf Sci 284:60–80MathSciNetCrossRefMATHGoogle Scholar
  30. Yao YY, Deng X (2014) Quantitative rough sets based on subsethood measures. Inf Sci 267:306–322MathSciNetCrossRefMATHGoogle Scholar
  31. Yao YY, Mi J, Li Z (2014) A novel variable precision (\(\theta \), \(\sigma \))-fuzzy rough set model based on fuzzy granules. Fuzzy Sets Syst 236:58–72MathSciNetCrossRefMATHGoogle Scholar
  32. Yao YY (2010) Three-way decisions with probabilistic rough sets. Inf Sci 180:341–353MathSciNetCrossRefGoogle Scholar
  33. Yin Y, Wang J (2010) Fuzzy Hemirings. Science Press, BeijingGoogle Scholar
  34. Yu B, Zhan J (2015) Applications of falling fuzzy \(h\)-bi-(\(h\)-quasi-)ideals to hemirings based on probability spaces. J Mult Valued Logic Soft Comput 25:237–267MathSciNetGoogle Scholar
  35. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefMATHGoogle Scholar
  36. Zhan J (2015) The uncertainties of ideal theory on hemirings. Science Press, BeijingGoogle Scholar
  37. Zhan J, Dudek WA (2007) Fuzzy \(h\)-ideals of hemirings. Inf Sci 177:876–886MathSciNetCrossRefMATHGoogle Scholar
  38. Zhan J, Liu Q, Davvaz B (2015) A new rough set theory: rough soft hemirings. J Intell Fuzzy Syst 28:1687–1697MathSciNetMATHGoogle Scholar
  39. Zhan J, Liu Q, Herawan T (2016) A novel soft rough set: soft rough hemirings and its multicriteria group decision making (in press) Google Scholar
  40. Zhan J, Shum KP (2015) Applications of soft union sets in \(h\)-hemiregular and \(h-intra\)-hemiregular hemirings. Bull Malays Math Sci Soc 38(2):805–825MathSciNetCrossRefMATHGoogle Scholar
  41. Zhang X, Dai J, Yu Y (2015) On the union and intersection operations of rough sets based on various approximation spaces. Inf Sci 292:214–229MathSciNetCrossRefMATHGoogle Scholar
  42. Zhu W (2009) Relationships among basic concepts in covering-based rough sets. Inf Sci 179:2479–2486MathSciNetGoogle Scholar
  43. Zhu W, Wang S (2013) Rough metroids based on relations. Inf Sci 232:241–252CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsHubei University for NationalitiesEnshiChina

Personalised recommendations